Varieties generated by free algebras of type (-1,1) of finite rank
Let \({\mathcal M}\) be the variety of (-1,1)-algebras over a field of characteristic 0, \({\mathcal M}_ n\) be the subvariety of \({\mathcal M}\), generated by a free algebra of rank n. It is well-known that \({\mathcal M}_ 1\subseteq {\mathcal M}_ 2\subseteq...\subseteq {\mathcal M}_ n\subseteq...\subseteq {\mathcal M}\), \({\mathcal M}=\cup_{n}{\mathcal M}_ n\). The author had shown earlier that \({\mathcal M}\neq {\mathcal M}_ n\) for any n [Algebra Logika 14, No.5, 543-571 (1975; Zbl 0346.17004)]. How he continues the study of the properties of the varieties \({\mathcal M}_ n\). Let A be a free (-1,1)-algebra of countable rank and \(A_ n\) its free subalgebra of rank n. If C is one of the centers K,N,Z (commutative, associative or associative-commutative center, correspondingly), then \(C^*\) denotes the largest ideal of an algebra B, contained in C(B), and \(C^*_ n=\{f\in A|\) \(f(A_ n)\subseteq C^*(A_ n)\}.\) It is proved that for any \(n\geq 1\) the inclusion \(C^*_ n\supset C^*_{n+1}\) is strong \((C=K,N,Z)\). In particular, for any n the inclusion \({\mathcal M}_ n\subseteq {\mathcal M}_{n+1}\) is also strong. Further, for any n the variety \({\mathcal M}_ n\) has a finite basis of identities and is represented as a union of the variety of all associative algebras and the variety generated by a certain finite- dimensional algebra.
scientific article; zbMATH DE number 4033969
scientific article